Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let X n , n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (1) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (01) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (3) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (03) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

scholarly journals Continuous, Discrete, and Conditional Scan Statistics

2012 ◽  
Vol 49 (01) ◽  
pp. 199-209 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu ◽  
W.Y. Wendy Lou
Keyword(s):  
Rates Of Convergence ◽  
Scan Statistics ◽  
Random Permutations ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Scan Statistic ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Runs And Patterns ◽  
Theoretical Results

The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.

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